Polymeric nature of tandemly repeated genes enhances assembly of constitutive heterochromatin in fission yeast

Motivated by our recent experiments that demonstrate that the tandemly repeated genes become heterochromatin, here we show a theory of heterochromatin assembly by taking into account the connectivity of these genes along the chromatin in the kinetic equations of small RNA production and histone methylation, which are the key biochemical reactions involved in the heterochromatin assembly. Our theory predicts that the polymeric nature of the tandemly repeated genes ensures the steady production of small RNAs because of the stable binding of nascent RNAs produced from the genes to RDRC/Dicers at the surface of nuclear membrane. This theory also predicts that the compaction of the tandemly repeated genes suppresses the production of small RNAs, consistent with our recent experiments. This theory can be extended to the small RNA-dependent gene silencing in higher organisms.

1. The authors motivate this work by contrasting this mechanism with phase separation. Specifically, they say that "The nucleus of a fission yeast has only three chromosomes, each having a centromeric region of 40 -110 kbps (which are estimated to be 24 -67 Kuhn units). Because phase separation is a collective phenomenon of many polymers, the heterochromatin of fission yeast is not likely to be assembled by the phase separation of chromatin." I would push back on this assertion. Even one chain can undergo phase separation if it is exceedingly long and highly multivalent. While the authors mention that the number of Kuhn segments is relatively small, the actual length of the chain is quite large -significantly larger than any proteins that undergo phase separation. Thus, I would suggest that the authors either include stronger evidence that these chains do not undergo phase separation or remove this statement altogether and state that they are suggesting an alternative mechanism of heterochromatin organization. 2. I am not an expert in heterochromatin organization, so please excuse my ignorance here. I found it slightly unclear exactly what the authors are proposing as the relationship between the tandem repeat nature of the centromeric genes and the binding of nascent RNAs to RDRC/Dicers. I recognize that they are drawing comparisons to surface adhesion of polymers as in Fig. 1, which makes sense. However, is it that the polymeric nature of the genes themselves promote binding to a specific surface? Or is the polymeric nature of the nascent RNAs promoting binding? Furthermore, is the surface in this case the surface of RDRC/Dicer molecules or the surface of the nuclear membrane? Based on the results of Figure 5, it seems that the genes are binding RDRC/Dicers, but I think these points could use some clarification early on for those less familiar with the prior work. I would suggest reworking Figure 2 to make the whole model clearer. 3. As a modeling paper, this work includes numerous symbols and parameters. I appreciate that the authors include two tables to describe the parameter values. However, I often found myself needing to look at prior paragraphs to recall what each symbol stands for. As such, I think it would be worthwhile to include a glossary that defines all of the unique symbols used throughout this work. This would be a useful reference for the reader. 4. In Figure 4, τ<sub>sp1</sub> and τ<sub>sp2</sub> should be defined in the figure caption. 5. Also in Figure 4, the magenta-colored dashed curve and dotted lines were hard to differentiate, which led to some difficulty in reading the graph at first. I recommend using a different method to demarcate τ<sub>sp1</sub> and τ<sub>sp2</sub>. 6. In Figure 5, the differently-colored curves typically overlay each other in the regime of long elongation times. Thus, saying that "The average degree σq<sub>on</sub> of H3K9 methylation and the average production rate of small RNAs also increase dramatically with increasing the number N of genes" seems misleading, since this is only true for certain values of the elongation time. 7. The legend for Table 2 states "The volume and concentration of Pol II are estimated to be 6 × 10<sup>3</sup> nm<sup>2</sup> (Spahr et al. 2009) and 50 (the number of Pol II per cell is 3 × 10<sup>4</sup> (Borggrefe et al. 2001) and the size of a nucleus of fission yeast is in the order of 1 μm<sup>3</sup> (Wang et al. 2016)), respectively." The volume units are nm<sup>2</sup>, which seems odd, and the concentration is unitless. I believe these both need to be fixed.

Reply to Reviewer #1
In this manuscript, the authors combine the approaches in systems biology and polymer physics to provide a modeling framework for the assembly of constitutive heterochromatin in fission yeast with considering the processes of small RNA production and histone methylation.
They reveal the role of the polymeric nature of tandemly repeated genes to enhance the assembly of constitutive heterochromatin. Their findings are of much importance to improve our understanding about the assembly of heterochromatin in cells. Therefore, I would like to recommend it for publication in Communications Biology after the following problems are addressed.
Thank you very much for your recommendation and constructive comments. Your comments were useful to make our manuscript more concise and to make it more accessible to wider audience. We thus revised our manuscript in line with your suggestions. The

It feels confused that '… is derived by the minimization of the free energy in the spirit of the Flory theory' without showing the minimizing of the free energy given by equation (M5).
Please add it or make other further explanations.
We summarized the derivation of eq. (9) from the minimization of the free energy (eq. (17)) in the Supplementary Note 3.
And more importantly, please explain why to use the free energy to obtain the volume fraction ¥phi_C and why it is reasonable and effective.
Thank you very much for this comment. The heterochromatin of fission yeasts is not in the thermodynamic equilibrium. It is therefore necessary to clearly explain why our approach is effective. First, the equation of state that relates the osmotic pressure and the volume fraction of chromatin is derived by the thermodynamic relationship. Our use of the free energy minimization by introducing the contribution of osmotic pressure is just for convenience. To avoid misunderstanding, we revised the sentence in L269-L275: Eq. (9) is an extension of the equation of state that predicts the coil-globule transition of polymers and is derived by using the free energy in the spirit of the Flory theory 50 , see 2 Second, to specify cases in which the balance of stresses in eq. (9) is applicable, we also added a sentence in the same paragraph: This treatment is effective for cases in which the binding and unbinding of the units is rate limited: in such cases, the polymer dynamics associated with the binding and unbinding of the units is negligible and thus the local equilibrium approximation is applicable in the length scale of the subchain. We added Supplementary Note 2 to show the derivation of Fig. 4, including two limits of instability, τ sp1 and sp2 , and the critical value c . We added a sentence in the caption of The derivation of this figure is shown in Supplementary Note 2.
We noticed that the labels of figs. 4 and 5 was wrong, rather than the text. We thus corrected these labels.
8. Please check the sentence 'Indeed, the production rate …, see Fig.6b and c' in Lines 321-

323.
To make the description more precise, we revised the corresponding sentence (L310-L312) as follows: Indeed, the production rate 2 of small RNAs and the degree σ of H3K9 methylation in the bound state decreases with increasing the interaction parameter for the regime of long elongation time, see Fig. 6b and c.
9. Please check the sentence 'The production rate of small RNAs … in Fig.7b and c' in Lines 338-341.
To make the description more precise, we revised the corresponding sentence (L325-L330) as follows: It implies that if the inverse demethylation rate dm −1 is larger than the maximum in the wild type, the binding probability p decreases with decreasing the rate dm (which corresponds to the depletion of Epe1). In this regime, the production rate of small RNAs in the bound state also decreases with decreasing the demethylation rate dm , while the degree of H3K9 methylation increases with decreasing the demethylation rate dm , see the orange and magenta lines in Fig. 7b and c.

It is better to add the meaning of z in equation (M9) in Line 529.
We added the following sentence at L518 below eq. (21) (eq. (M9) in the previous version) to address the meaning of : is the coordinate from the surface of the nuclear membrane, see Fig. 2.

I feel confuse for the orange color of RDRC/Dicers used in Fig.2a. And, it is better to
indicate what a promoter is in Fig.2b.
We changed the color of RDRC/Dicers in Fig. 2a to black and indicated promoter in Fig. 2b.

Reply to Reviewer #2
In their paper "Polymeric nature of tandemly repeated genes enhances assembly of constitutive heterochromatin in fission yeast," Yamamoto, Asanuma, and Murakami develop a kinetic model to understand the relationship between heterochromatin assembly, small RNA production, and histone methylation in fission yeast. This paper is a follow-up to an experimental paper published by many of the same authors. As I understand them, the main results and predictions set forth by this paper are as follows: Thank you very much for your recommendation and constructive comments. Your comments are useful to make our manuscript more concise and make it accessible to wider audience. We thus revised our manuscript in line with your suggestions. The revised parts are highlighted by red letters. Our point-by-point reply follows: 1. The authors motivate this work by contrasting this mechanism with phase separation.

Specifically, they say that "The nucleus of a fission yeast has only three chromosomes, each having a centromeric region of 40 -110 kbps (which are estimated to be 24 -67 Kuhn units).
Because phase separation is a collective phenomenon of many polymers, the heterochromatin of fission yeast is not likely to be assembled by the phase separation of chromatin." I would push back on this assertion. Even one chain can undergo phase separation if it is exceedingly long and highly multivalent. While the authors mention that the number of Kuhn segments is relatively small, the actual length of the chain is quite largesignificantly larger than any proteins that undergo phase separation. Thus, I would suggest that the authors either include stronger evidence that these chains do not undergo phase separation or remove this statement altogether and state that they are suggesting an alternative mechanism of heterochromatin organization.
In line with your suggestion, we removed the statement altogether from both abstract and introduction. reworking Figure 2 to make the whole model clearer.

I am not an expert in heterochromatin organization
We added thick cyan and black lines that highlight the connectivity of genes along chromatin and the `connectivity' due to the localization of RDRC/Dicers at the surface of nuclear membrane in Figure 2: We also added the following sentence (L706-L709) in the caption of Fig. 2 to describe the meaning of these thick lines: If more than one gene are bound to a RDRC/Dicer, unbound genes in the tandem repeat are also localized at the vicinity of RDRC/Dicers because the genes are connected through the chromatin and RDRC/Dicers are localized at the surface of the nuclear membrane, see the thick cyan and black lines.
To avoid mistake between the surfaces of nuclear membranes and the surfaces RDRC/Dicers, we make sure to write "the surface of the nuclear membrane", not just "surface". To avoid mistake between the polymeric nature of chromatin and the polymeric nature of nascent RNAs, we added the following sentence at L118-L121: Nascent RNAs produced by the transcription of heterochromatin regions are retained to the chromatin via RITS complexes 28,42,43 . A complex of the chromatin unit and nascent RNA can be thus viewed as one unit that can bind to RDRC/Dicers at the surface of the nuclear membrane.
We also revised Fig. 2 to make clear that nascent RNAs are retained to chromatin by RITS complexes (if the chromatin is H3K9 methylated).
To address the punchline of this paper clearly, we also revised a sentence on L395-L400: The stable binding is promoted by the localization of nascent RNAs along DNA and of RDRC/Dicers on the surface of the nuclear membrane with a mechanism analogous to the surface adhesion of polymers: if a nascent RNA produced from a gene is bound to a RDRC/Dicer, other nascent RNAs in the tandem repeat are at the vicinity to the surface, where other RDRC/Dicers are localized. Thank you for this idea. We added a glossary of symbols in Supplementary Table 2. Figure 4, τsp1 and τsp2 should be defined in the figure caption.

In
To clarify the definition of sp1 and τ sp2 , we added the following sentence in the caption of  Figure 4, the magenta-colored dashed curve and dotted lines were hard to differentiate, which led to some difficulty in reading the graph at first. I recommend using a different method to demarcate τsp1 and τsp2.

Also in
We used green color (instead of magenta) dotted lines to demarcate sp1 and τ sp2 .